Properties of set operators Commutativety: AnB = BnA and AUB = BUA. Associativety: An(BOC) = (AnB)nC and AU (BUC) = (AUB) UC. Distributive laws: An (BUC) = (An B) U (AnC) and: AU(BOC) = (A UB) n (A UC). De Morgan's laws: (AnB) = AUB and (AUB) = AnBc. Further properties of set operators If S is the sample space and A and B are any sets in S, you can also use the following results without proof: Oc = S. O CA, A C A and A C S. . An A = A and A U A = A. An Ac = 0 and A U Ac = S. If B C A, AnB = B and A UB = A. . An0 = 0 and A UO = A. AnS = A and A US = S. - 0n0 = 0 and QUO = 0.(a) Use the rules of set operators (on page 29) to prove that the following represents a partition of set A: A:(AB)U(ABC). (3) In other words, prove that (3) is true, and also that (A O B) D (A H BC) : ill. (b) The following are also partitions: B:(BAC)U(BA) (4) AUB:(AHBC)U(AHB)U(ACHB). (5) Using the result that (3), (4) and (5) are partitions, prove the probability result: P(A o B): P(A) + 13(3) P(A n B)